266 6. Separation of Electronic and Nuclear Motions Fig. 6.15. The photochemical funnel effect. We can see two adiabatic surfaces (upper and lower), which resulted from intersection of two diabatic surfaces (white and gray). The lower surface corresponds to an electronic ground state, the upper to an excited electronic state. The molecule is excited from its ground state at the nuclear configuration P  to the excited state (point FC) at the same nuclear configuration (according to the Franck–Condon rule). The point FC representing the system is usually located on a slope of the potential energy (corresponding to the excited state) and this is why it slides downhill towards the energy minimum M ∗ . Its kinetic energy may be sufficient to go through M ∗ and pass a barrier (saddle point) corresponding to the point TS. Then, the system inevitably slides down into the conical intersection point C (“funnel effect”) and lands in the ground state surface (at the configuration of the conical intersection) with nearly 100% efficiency. The future of the system may correspond to different products: it may roll down to product P or slide back to product P  . Modified and adapted from F. Bernardi, M. Olivucci, M.A. Robb, Chem.Soc.Rev. (1996) 321. The electronic excitation takes place so fast that the nuclei do not have enough time to move. Thus the positions of the nuclei in the excited state are identical to those in the ground state (Franck–Condon rule). The point FC in Fig. 6.15 shows the very essence of the Franck–Condon rule – a vertical transition. The corresponding nuclear configuration may differ quite sig- vertical transition nificantly from the nearest potential energy minimum M ∗ in the excited state PES (E − ). In a few or a few tens of femtoseconds, the system slides down from P  to the 6.12 Polyatomic molecules and conical intersection 267 neighbourhood of M ∗ , transforming its potential energy into kinetic energy. Usu- ally point M ∗ is separated from the conical intersection configuration C by a bar- rier with the corresponding potential energy saddle point TS (“transition state”). Behind the saddle point there is usually an energy valley 57 with a deep funnel end- ing in the conical intersection configuration. As soon as the system overcomes the barrier at TS, by going over it or by tunnelling, it will be sucked in by the conical intersection attractor with almost 100% probability. funnel effect The system goes through the “funnel” to the electronic ground-state hyper- surface. Then the system will continue its path in the ground state PES, E + .Ifitsmo- mentum is large enough, the system slides down along path P towards the nearest local minimum. If its momentum is small, the system may choose path P  .TheP trajectory may mean a new reaction product, while P  means returning to the orig- inal molecule. Of course, the total energy has to be conserved. The non-radiative process de- scribed will take place if the system finds a way to dissipate its energy, i.e. to trans- energy dissipation fer an excess of electronic energy into the vibrational, rotational and translational degrees of freedom of its own or neighbouring molecules (e.g., of the solvent). 58 What will the energy in the plane ξ 1 ξ 2 be, far away from the conical intersection point? Of course, there is no reason for the energy to change linearly. Instead we may expect a more complex landscape to emerge on the E + and E − PESs, such as minima, saddle points, etc. shown in Fig. 6.15. We may ask whether we will find some other conical intersections in the ground-state PES. In general the answer is positive. There are at least two reasons for this. In the simplest case the conical intersection represents the dilemma of an atom C (approaching molecule AB): to attach either to A or B? Thus any encounter of three atoms causes a conical intersection (see Chap- ter 14). In each case the important thing is a configuration of nuclei, where a small variation may lead to distinct sets of chemical bonds like in an equilateral trian- gle configuration of H 3 . Similar “pivot points” may happen for four, five, six etc. atoms. Thus we will encounter not only the minima, maxima and saddle points, but also the conical intersection points when travelling in the ground-state PES. The second reason is the permutational symmetry. Very often the system con- tains the same kinds of nuclei. Any exchange of the positions of such nuclei moves the point representing the system in configuration space to some distant regions, whereas the energy does not change at all. Therefore, any PES has to exhibit the 57 OntheexcitedstatePES. 58 The energy is usually distributed among the degrees of freedom in an unequal way. 268 6. Separation of Electronic and Nuclear Motions corresponding permutational symmetry. All the details of PES will be repeated N! times for a system with N identical nuclei. This will multiply the number of conical intersections. More about conical intersection will be given in Chapter 14, when we will be equipped with the theoretical tools to describe how the electronic structure changes during chemical reactions. 6.13 BEYOND THE ADIABATIC APPROXIMATION. . . 6.13.1 MUON CATALYZED NUCLEAR FUSION Andrei Dmitrievich Sakharov (1921–1989) Russian physi- cist, father of the Soviet hy- drogen bomb. During the fi- nal celebration of the H bomb project Sakharov expressed his hope that the bombs would never be used. A Soviet gen- eral answered coldly that it was not the scientists’ busi- ness to decide such things. This was a turning point for Sakharov and he began his fight against the totalitarian system. The idea of muon induced fusion was conceived by Sa- kharov in 1945, in his first sci- entific paper, under the su- pervision of Tamm. In 1957 David Jackson realized that muons may serve as cata- lysts. Some molecules look really peculiar, they may contain a muon instead of an electron. The muon is an unstable parti- cle with the charge of an electron and mass equal to 207 electron masses. 59 For such a mass, assuming that nuclei are infinitely heavier than a muon looks like a very bad approximation. There- fore, the calculations need to be non- adiabatic. The first computations for muonic molecules were performed by Kołos, Roothaan and Sack in 1960. 60 The idea behind the project was muon catalyzed fusion of deuterium and tri- tium. This fascinating problem was pro- posed by the Russian physicist Andrei Sakharov. Its essence is as follows. If an electron in the molecule dt + is replaced by a muon, immediately the di- mension of the molecule decreases by a factor of about 200. How is this possible? The radius of the first Bohr orbit in the hydrogen atom (see p. 179) is equal to a 0 = ¯ h 2 μe 2 . After introducing atomic units, this formula becomes a 0 = 1 μ ,andwhen we take into account that the reduced mass μ ≈ m (m stands for the electron mass) we get a 0 ≈1. This approximation works for the electron, because in reality μ =09995m. If, in the hydrogen atom, instead an electron we have a muon, then μ would be equal about 250m. This, however, means that such a “muon Bohr radius” would be about 250 times smaller. Nuclear forces begin to operate at such a small muon catalysis internuclear separation (strong interactions, Fig. 6.16.a), and are able to overcome 59 The muon was discovered in 1937 by C.D. Anderson and S.H. Neddermeyer. Its life time is about 22·10 −6 s. The muons belong to the leptons family (with the electron and τ particle, the later with mass equal to about 3640 electron masses). Nature created, for some unknown reasons, the “more massive electrons”. When the nuclear physicist Isidor Rabi was told about the incredible mass of the τ particle, he dramatically shouted: “Who ordered that?!” 60 W. Kołos, C.C.J. Roothaan, R.A. Sack, Rev. Mod. Phys. 32 (1960) 205. 6.13 Beyond the adiabatic approximation. 269 Fig. 6.16. (a) The interaction energy potential (E)ofd and t as a function of the interparticle dis- tance (R), with taking the nuclear forces into account (an outline). At large R, of the order of nanome- ters, we have Coulombic repulsion, at distances of the order of femtometers the internuclear attractive forces (called the strong interaction) are switched on and overcome the Coulombic repulsion. At a dis- tance of a fraction of femtometer again we have a repulsion (b) “Russian dolls” (outline): the analogues of H 2 and H + 2 . the Coulombic barrier and stick the nuclei together by nuclear fusion.Themuon, however, is released, and may serve as a catalyst in the next nuclear reaction. Deuteron and tritium bound together represent a helium nucleus. One muon may participate in about 200–300 such muon catalyzed fusion processes. 61 Every- body knows how much effort and money has been spent for decades (for the moment without success) to ignite the nuclear synthesis d + t → He. Muon cat- alyzed fusion might be an alternative solution. If the muon project were success- ful, humanity would have access to a practically unlimited source of energy. Un- fortunately, theoretical investigations suggest that the experimental yield already achieved is about the maximum theoretical value. 62 61 The commercial viability of this process will not be an option unless we can demonstrate 900 fusion events for each muon. About 10 grams of deuterium and 15 g of tritium fusion would then be sufficient to supply the average person with electricity for life. 62 More about this may be found in K. Szalewicz, S. Alexander, P. Froelich, S. Haywood, B. Jeziorski, W. Kołos, H.J. Monkhorst, A. Scrinzi, C. Stodden, A. Velenik, X. Zhao, in “Muon Catalyzed Fusion”, eds. S.E. Jones, J. Rafelski, H.J. Monkhorst, AIP Conference Proceedings 181 (1989) 254. 270 6. Separation of Electronic and Nuclear Motions 6.13.2 “RUSSIAN DOLLS” – OR A MOLECULE WITHIN MOLECULE Scrinzi and Szalewicz 63 carried out non-adiabatic calculations (p. 224) for a sys- tem of 6 particles: proton (p), deuterium (d), tritium (t), muon (μ)andtwoelec- trons (e) interacting by Coulombic forces (i.e. no nuclear forces assumed). It is not so easy to predict the structure of the system. It turns out that the resulting struc- ture is a kind of “Russian doll” 64 (Fig. 6.16.b): the muon has acted according to its mass (see above) and created tdμ with a dimension of about 002 Å. This system maybeviewedasapartlysplitnucleusofcharge+1 or, alternatively, as a mini model of the hydrogen molecular ion (scaled at 1:200). The “nucleus” serves as a partner to the proton and both create a system similar to the hydrogen molecule, in which the two electrons play their usual binding role, and the internuclear distance is about 0.7 Å. It turns out that the non-zero dimension of the “nucleus” makes a difference, and the energies computed with and without an approximation of the point-like nucleus differ. The difference is tiny (about 0.20 meV), but it is there. It is quite remarkable that such small effects are responsible for the fate of the total system. The authors observe that the relaxation of the “nucleus” dtμ (from the excited state to the ground state 65 ) causes the ionization of the system: one of the electrons flies off. Such an effect may excite those who study this phenomenon. How is it possible? The “nucleus” is terribly small when seen by an electron orbit- ing far away. How could the electron detect that the nucleus has changed its state and that it has no future in the molecule? Here, however, our intuition fails. For the electron, the most frequently visited regions of the molecule are the nuclei. We will see this in Chapter 8, but even the 1s state of the hydrogen atom (p. 178, the maximum of the orbital is at the nucleus) suggests the same. Therefore, no wonder the electron could recognize that something has abruptly changed on one of the nuclei and (being already excited) it felt it was receiving much more freedom, so much that it could leave the molecule completely. We may pose an interesting question, whether the “Russian doll” represents the global minimum of the particles system. We may imagine that the proton changes its position with the deuterium or tritium, i.e. new isomers (isotopomers 66 )appear. 63 A. Scrinzi, K. Szalewicz, Phys. Rev. A 39 (1989) 4983. 64 (((woman@ woman)@ woman)@) 65 A. Scrinzi, K. Szalewicz, Phys.Rev.A39 (1989) 2855. The dtμ ion is created in the rovibrational state J = 1, v =1, and then the system spontaneously goes to the lower energy 01 or 00 states. The energy excess causes one electron to leave the system (ionization). This is an analogue of the Auger effect in spectroscopy. 66 The situation is quite typical, although we very rarely think this way. Some people say that they observe two different systems, whereas others say, that they see two states of the same system.Thisbegins with the hydrogen atom – it looks different in its 1s and 3p z states. We can easily distinguish two different conformations of cyclohexane, two isomers of butane, and some chemists would say these are different substances. Going much further, N 2 and CO represent two different molecules, or is one of them nothing but an excited state of the other? However strange it may sound for a chemist, N 2 represents an excited state of CO, because we may imagine a nuclear reaction of the displacement of a proton from one nitrogen to the other (and the energy curve per nucleon as a function of the atomic Summary 271 The authors did not study this question, 67 but investigated a substitution of the proton by deuterium and tritium (and obtained similar results). Scrinzi and Szalewicz also performed some calculations for an analogue of H + 2 : proton, deuterium, tritium, muon and electron. Here the “Russian doll” looks won- derful (Fig. 6.16.c); it is a four-level object: • the molecular ion (the analogue of H + 2 )iscomposedofthree objects: the proton, the “split nucleus” of charge +1 and the electron; • the “split nucleus” is also composed of three objects: d, t, μ (a mini model of H + 2 ); • the tritium is composed of three nucleons: the proton and the two neutrons; • each of the nucleons is composed of three quarks (called the valence quarks). Summary • In the adiabatic approximation, the total wave function is approximated as a product  = ψ k (r;R)f k (R) of the function f k (R), which describes the motion of the nuclei (vibrations and rotations) and the function ψ k (r;R) that pertains to the motion of electrons (and depends parametrically on the configuration of the nuclei; here we give the formulae for a diatomic molecule). This approximation relies on the fact that the nuclei are thousands of times heavier than the electrons. • The function ψ k (r;R) represents an eigenfunction of the Hamiltonian ˆ H 0 (R) of eq. (6.4), i.e. the Hamiltonian ˆ H, in which the kinetic energy operator for the nuclei isassumedtobezero(theclamped nuclei Hamiltonian). • The function f k (R) is a product of a spherical harmonic 68 Y M J that describes the rotations of the molecule (J and M stand for the corresponding quantum numbers) and a function that describes the vibrations of the nuclei. • The diagram of the energy levels shown in Fig. 6.3 represents the basis of molecular spectroscopy. The diagram may be summarized in the following way: – the energy levels form some series separated by energy gaps, with no discrete levels. Each series corresponds to a single electronic state n, and the individual levels pertain to various vibrational and rotational states of the molecule in electronic state n; – within the series for a given electronic state, there are groups of energy levels, each group characterized by a distinct vibrational quantum number (v = 0 1 2),and within the group the states of higher and higher energies correspond to the increasing rotational quantum number J; – the energy levels fulfil some general relations: ∗ increasing n corresponds to an electronic excitation of the molecule (UV-VIS, ultra- violet and visible spectrum), ∗ increasing v pertains to a vibrational excitation of the molecule, and requires the energy to be smaller by one or two orders of magnitude than an electronic excitation (IR, infrared spectrum). ∗ increasing J is associated with energy smaller by one or two orders of magnitude than a vibrational excitation (microwaves). mass is convex). Such a point of view is better for viewing each object as a “new animal”: it enables us to see and use some relations among these animals. 67 They focused their attention on tdμ. 68 That is, of the eigenfunction for the rigid rotator. 272 6. Separation of Electronic and Nuclear Motions • Theelectronicwavefunctionsψ k (r;R) correspond to the energy eigenstates E 0 k (R), which are functions of R. The energy curves 69 E 0 k (R) for different electronic states k may cross each other, unless the molecule is diatomic and the two electronic states have thesamesymmetry. 70 In such a case we have what is known as an avoided crossing (see Fig. 6.12). • For polyatomic molecules the energy hypersurfaces E 0 k (R) can cross. The most important is conical intersection (Fig. 6.15) of the two (I and II) diabatic hypersurfaces, i.e. those that (each individually) preserve a given pattern of chemical bonds. This intersection results in two adiabatic hypersurfaces (“lower and upper”). Each of the adiabatic hypersurfaces consists of two parts: one belonging to I and the second to II. Using a suitable coordinate system in the configurational space, we obtain, independence of the adiabatic hypersurface splitting of 3N − 8 coordinates and dependence on two coordinates (ξ 1 and ξ 2 ) only.The splitting begins by a linear dependence on ξ 1 and ξ 2 , which gives a sort of cone (hence the name “conical intersection”). • Conical intersection plays a prominent role in the photochemical reactions, because the excited molecule slides down the upper adiabatic hypersurface to the funnel (just the con- ical intersection point) and then, with a yield close to 100% lands on the lower adiabatic hypersurface (assuming there is a mechanism for dissipation of the excess energy). Main concepts, new terms clamped nuclei Hamiltonian (p. 223) non-adiabatic theory (p. 224) adiabatic approximation (p. 227) diagonal correction for the motion of the nuclei (p. 227) Born–Oppenheimer approximation (p. 229) potential energy curve (p. 231) potential energy (hyper)surface (p. 233) electronic-vibrational-rotational spectroscopy (p. 235) non-bound states (p. 247) non-bound metastable states (p. 247) wave function “measurement” (p. 251) diabatic curve (p. 253) adiabatic curve (p. 253) avoided crossing (p. 255) non-crossing rule (p. 256) harpooning effect (p. 257) conical intersection (p. 262) Berry phase (p. 264) Franck–Condon rule (p. 266) funnel effect (p. 266) non-radiative transitions (p. 266) photochemical reaction (p. 266) muon catalyzed nuclear fusion (p. 268) split nucleus effect (p. 270) From the research front Computing a reliable hypersurface of the potential energy (PES) for the motion of nuclei, represents an extremely difficult task for today’s computers, even for systems of four atoms. In principle routine calculations are currently performed for three-atomic (and, of course, two-atomic) systems. The technical possibilities are discussed by J. Hinze, A. Alijah and L. Wolniewicz, Pol. J. Chem. 72 (1998) 1293, in which the most accurate calculations are also reported (for the H + 3 system). Analysis of conical intersections is only occasionally carried out, because the problem pertains to mostly unexplored electronic excited states. 69 As functions of R. 70 That is, they transform according to the same irreducible representation. Ad futurum. 273 Ad futurum. . . The computational effort needed to calculate the PES for an N atomic molecule is pro- portional to 10 3N−6 . This strong dependence suggests that, for the next 20 years, it would be rather unrealistic to expect high-quality PES computations for N>10. However, ex- perimental chemistry offers high-precision results for molecules with hundreds of atoms. It seems inevitable that it will be possible to freeze the coordinates of many atoms. There are good reasons for such an approach, because indeed most atoms play the role of specta- tors in chemical processes. It may be that limiting ourselves to, say, 10 atoms will make the computation of rovibrational spectra feasible. Additional literature J. Hinze, A. Alijah, L. Wolniewicz, “Understanding the Adiabatic Approximation; the Accurate Data of H 2 Transferred to H + 3 ”, Pol. J. Chem. 72 (1998) 1293. The paper reports the derivation of the equation of motion for a polyatomic molecule. As the origin of the BFCS, unlike this chapter, the centre of mass was chosen. 71 W. Kołos, “Adiabatic Approximation and its Accuracy”, Advan. Quantum Chem. 5 (1970) 99. Kołos was the No 1 expert in the domain. F. Bernardi, M. Olivucci, M.A. Robb, “Potential Energy Surface Crossings in Organic Photochemistry”, Chem. Soc. Rev. p. 321–328 (1996). A review article by the top experts in conical intersection problems. Questions 1. A diatomic homonuclear molecule, origin of the BFCS in the centre of the molecule, po- tential energy of the Coulombic interactions equals V . The total non-relativistic Hamil- tonian is equal to: a) ˆ H =−  i ¯ h 2 2m  i − ¯ h 2 2μ  R ;b) ˆ H =−  i ¯ h 2 2m  i + V ;c) ˆ H =− ¯ h 2 2μ  R + V ;d) ˆ H = −  i ¯ h 2 2m  i +V − ¯ h 2 2μ  R − ¯ h 2 8μ (  i ∇ i ) 2 . 2. A diatomic molecule, origin of the BFCS in the centre of the molecule. In the adiabatic approximation, the total wave function is in the form  =ψ k (r;R)f k (R),where: a) f k (R) describes the translation of the molecule; b) f k (R) stands for a spherical har- monic describing the rotations of the molecule; c) ψ k (r;R) denotes the eigenfunction of the clamped nuclei Hamiltonian; d) ψ k (r;R) stands for the probability density of having the electrons with coordinates r and the nuclei at distance R. 3. A diatomic molecule, origin of the BFCS in the centre of the molecule, in the adiabatic approximation the total wave function is in the form  =ψ k (r;R)f k (R) The potential energy for the vibrations of the molecule is equal to: a) V + J(J + 1) ¯ h 2 2μR 2 ;b)ψ k | ˆ Hψ k +(2J + 1) ¯ h 2 2μR 2 ;c)ψ k | ˆ Hψ k +J(J + 1) ¯ h 2 2μ ; d) ψ k | ˆ Hψ k +J(J +1) ¯ h 2 2μR 2 . 71 We have chosen the centre of the ab bond. 274 6. Separation of Electronic and Nuclear Motions 4. The potential energy curves for the motion of the nuclei for a diatomic molecule a) cross, if their derivatives differ widely; b) always cross; c) do not cross, if they corre- spond to the same irreducible representation of the symmetry group of the Hamiltonian; d) do not cross, if they correspond to different symmetry. 5. Please choose the wrong answer. The potential energy for the motion of the nuclei: a) contains the eigenvalue of the clamped nuclei Hamiltonian; b) does not change when the rotation excitation occurs; c) represents the electronic energy when the Born–Oppenheimer approximation is used; d) for bound states has to be convex as a function of the positions of the nuclei. 6. Please choose the wrong answer. As a result of the rotational excitation J → (J +1) of a molecule of length R: a) the angular momentum increases; b) the potential for vibrations changes; c) the potential energy curve for the motion of the nuclei becomes shallower; d) the potential energy increases by a term proportional to (2J + 1) and proportional to R −2 . 7. The potential energy hypersurface for the N-atomic molecule depends on the following number of variables: a) 2N −8; b) 3N −6; c) 3N −5; d) N. 8. At the conical intersection (Born–Oppenheimer approximation), the cone angle: a) does not depend on the direction of leaving the conical intersection point; b) is dif- ferent for the lower and for the higher cones; c) depends on the values of coordinates other than those along directions ∇( ¯ E 1 − ¯ E 2 ) and ∇(V 12 ); d) is different for different isotopomers. 9. At the conical intersection the following directions in configurational space lead to split- ting between E + and E − : a) ∇( ¯ E 1 − ¯ E 2 ) and ∇(V 12 );b)∇( ¯ E 1 ) and ∇( ¯ E 2 );c)∇( ¯ E 1 · ¯ E 2 ) and ∇(V 12 );d)∇( ¯ E 1 + ¯ E 2 ) and ∇(V 12 ). 10. Please find the wrong answer. The adiabatic approximation: a) is equivalent to the Born–Oppenheimer approximation; b) is related to the wave func- tion in the form of a product of an electronic function and a function describing the motion of the nuclei; c) leads to the notion of the potential energy curve for the motion of the nuclei; d) is worse satisfied for molecules with muons instead of electrons. Answers 1d, 2c, 3d, 4c, 5b, 6d, 7b, 8c, 9a, 10a Chapter 7 MOTION OF NUCLEI Where are we? We are on the most important side branch of the TREE. An example Which of conformations (Fig. 7.1) is more stable: the “boat” or “chair” of cyclohexane C 6 H 12 ? How do particular conformations look in detail (symmetry, interatomic distances, bond angles), when the electronic energy as a function of the positions of the nuclei attains a minimum value? What will be the most stable conformation of the trimer: C 6 H 11 –(CH 2 ) 3 – C 6 H 10 –(CH 2 ) 3 –C 6 H 11 ? Fig. 7.1. The chair (a) and boat (b) conformations of cyclohexane. These geometries (obtained from arbitrary starting conformations) are optimized in the force field, which we will define in the present chapter. The force field indicates, in accordance with experimental results, that the chair conformation is the more stable (by about 5.9 kcal/mol). Thus we obtain all the details of the atomic positions (bond lengths, bond angles, etc.). Note that the chair conformation obtained exhibits D 3d symmetry, while the boat conformation corresponds to D 2 (the boat has somewhat warped planks, because of repulsion of the two upper hydrogen atoms). What is it all about Rovibrational spectra – an example of accurate calculations: atom–diatomic molecule () p. 278 • Coordinate system and Hamiltonian 275 . analogue of H + 2 )iscomposedofthree objects: the proton, the “split nucleus” of charge +1 and the electron; • the “split nucleus” is also composed of three objects: d, t, μ (a mini model of H + 2 ); •. has to be convex as a function of the positions of the nuclei. 6. Please choose the wrong answer. As a result of the rotational excitation J → (J +1) of a molecule of length R: a) the angular momentum. func- tion in the form of a product of an electronic function and a function describing the motion of the nuclei; c) leads to the notion of the potential energy curve for the motion of the nuclei; d)